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\begin{document}

\centerline {\bf Math 523b - Homework 5} 

\bigskip

\begin{enumerate}

\item Folland problem 19, p. 192

\item Folland problem 20a, p. 192. Note that only part (a) is assigned.

\item Folland problem 32, p. 197

\item Folland problem 36, p. 199

\item Folland problem 39, p. 199

\item Folland problem 2, p. 215

\item Folland problem 4, p. 215

\item 
This problem gives an example to show that the measure
associated with a postive linear functional on $C_c(X)$ by
the Riesz representation theorem need not be regular. 
Let $X$ be the plane with the following topology. A set is open
if and only if its intersection with every vertical line is an 
open subset of that line. Show that $X$ is a locally compact Hausdorff
space. Let $f \in C_c(X)$. Show that there are only finitely
many $x$ such that $f(x,y) \ne 0$ for at least one $y$. 
Let $x_1,x_2,\cdots x_n$ be these $x$'s. Define 
$$F(f) = \sum_{j=1}^n \, \int_{- \infty}^\infty \, f(x_j,y) dy$$
Obviously $F$ is a positive linear functional, and so is represented
by a measure $\mu$. Let $E$ be the $x$-axis. Show that 
$\mu(E)= \infty$ but $\mu(K)=0$ for every compact $K \subset E$.




\end{enumerate}
\end{document}
\end













